Strings and Things for Locating Earthquakes
D. Sarah Stamps and Robert Smalley, Jr.
D. Sarah Stamps and Robert Smalley, Jr.
Center for Earthquake Research and Information, The University of Memphis
INTRODUCTION
When one hears that an earthquake has occurred, one of the first
questions is, “Where was it?” For the general public, this question
often determines the importance of another question, “How big
was it?” Upon learning the location and size, one may wonder
how this information was determined. For those interested in
how earthquakes are located, we have developed an interactive,
three-dimensional analog computer that uses a map, strings,
and a time-distance scale to find the location of an earthquake
(latitude, longitude, and depth) based on seismic wave arrival
times. We also have developed a set of lesson plans to present
the ideas used for locating earthquakes to grade-school students,
college classes, and the general public. The device is suitable for
both permanent mounting in a science museum or can be transported easily to schools or other educational and outreach venues. Our prototype analog earthquake locator is housed in the
Public Earthquake Resource Center (PERC) at the University
of Memphis and also is taken to local classrooms.
The most popular introductory method for showing how
to locate earthquakes is the circle method, which is based on
seismic
station
r = v(Ts – Tp)
epicenter
▲ Figure 1. This figure illustrates the circle method for locating
earthquakes. First draw circles of appropriate radii, based on the
Ts – Tp times, around each station on a map. Data from one station
locates the earthquake somewhere on the circle drawn about that
station. Data from two stations further restricts the earthquake
location to one of the two places where the two circles intersect.
Finally, the intersection of the three circles gives the unique location of the earthquake.
drawing circles around seismic stations on a map or a globe (figure 1). The circles’ radii are based on the length of time between
the arrival of the P and S waves, Ts – Tp , at each station. This
time difference is converted to a distance by using a table or
multiplying by a given velocity. How the velocity or the values
in the table are determined is usually not discussed. The distance
determined from the Ts – Tp time is used as the radius of a circle
that can be drawn on the map or globe by a student using a compass. Laboratory exercises and presentations about earthquakes
to elementary, middle, and high-school groups typically apply
this method with a compass, a regional map, and data from at
least three stations. In this exercise, the distance between the
earthquake and the seismic stations has to be small enough that
the Earth can be considered flat. Drawing circles with appropriate radii around the seismic stations produces a point where the
three circles intersect. The intersection point is the surface location, or epicenter, of the earthquake. This demonstration works
perfectly for earthquakes at the surface and noiseless P- and Swave arrival time data. Real earthquakes, however, occur inside
the Earth at some depth, not at the surface. In this case, the
circles drawn on the map will not intersect at one point because
the distances determined using the Ts – Tp arrival times are the
distances from the earthquake to the stations, not the distances
from the point on the Earth’s surface above the earthquake to
the stations (figure 2). The problem can be fixed by using the distances found from the Ts – Tp arrival times as the radii of spheres
in three dimensions, instead of two-dimensional circles on the
surface. The three radii now create two points of intersection.
Both points are equidistant to the Earth’s surface. One is located
within the Earth and the other above the Earth’s surface. The
point of intersection beneath the surface is the focal location of
the earthquake; hence the distance from the surface to the point
of intersection above the Earth’s surface also denotes the depth
of the earthquake. The radii represent the ray paths of the seismic rays traveling through a homogeneous medium. The analog
earthquake locator forms the intersection of the radii of spheres
in three dimensions above the surface of a map, demonstrating
not only the epicentral location on the surface directly beneath
the intersection, but the depth of the earthquake as well.
ANALOG COMPUTING
The term “analog” has two meanings that are relevant to the
earthquake locator. The fundamental principle behind the
application of analog computing is the observation that many
seemingly different physical systems can be described mathematically by equations of the same form, differing only in the
interpretation of the parameters and variables. If two systems
Seismological Research Letters Volume 77, Number 6 November/December 2006 677
hypocenter
r = v (Ts – Tp)
d
epicenter
seismic
station
circles
on surface
▲ Figure 2. This figure illustrates the failure of the circle method when the earthquake does not occur at the surface. The three radii are
shown intersecting at a single point, the hypocenter, but this point is not on the surface. The failure of the circle method gets more pronounced as the depth of the earthquake becomes equal to or greater than the epicentral distance (the distance from the epicenter, on the
surface, to the seismic stations). The distance the seismic waves travel between the earthquake and the stations, given by Ts – Tp , determines
the radii of spheres about the hypocenter. Because the waves travel the distances shown by the heavy lines, when the radii are rotated to
make circles on the surface (fine lines on surface), the radii are all too long and the three circles do not intersect at a single point.
are equivalent mathematically, we can investigate the behavior
of one system in terms of the other. Electrical and plumbing circuits are an example of analogous systems. In these two systems
the mathematical relationships between current and fluid flow,
and voltage and pressure, are the same. One could therefore use
electric circuits to simulate plumbing systems and determine the
pressure, current, and resistance for a given plumbing system by
making measurements on the analogous electrical circuit. The
second meaning refers to the use of continuous, as opposed to
digital, quantities. Analog computers use continuously varying
quantities to perform their calculations, although the actual calculation may depend on discrete rather than continuous levels.
Analog computers have had numerous applications.
Electrical analog computers were once a popular method for
solving large differential equations. Another example of an analog computer is the simple slide rule. There are an incredible
number of interesting nonelectrical analog computing devices,
such as the famous Norden bombsight of World War II. At
one time the U.S. Geological Survey used the circle method
to locate teleseisms by finding the intersection of circles on a
globe. Although analog systems have been replaced by digital computers, the earthquake locator is a useful, appropriate
device because its purpose is education rather than speed.
THE ANALOG EARTHQUAKE LOCATOR
The analog computer method for locating earthquakes provides students and visitors of the PERC at the Center for
Research Information (CERI) a hands-on visual presentation to help understand how to determine earthquake focal
locations in three dimensions. It also introduces some of the
physical ideas, such as ray paths, associated with this process.
The locator was inspired by a device built and used by the late
Argentine seismologist Fernando Volponi, of the Instituto
Geofísico Sismológico Zonda (now Instituto Sismológico
“Fernando Volponi”) of the Universidad Nacional de San Juan
in San Juan, Argentina. I. Selwyn Sacks suggested the device
to Volponi to address the issue of locating intermediate-depth
earthquakes beneath San Juan, where the circle method fails
due to the 100+ km depth of earthquakes; epicentral distances
for events immediately beneath the network were often much
smaller than their depths. The size of our device is scaled for
determining the epicenter and depth for earthquakes within a
small part of the New Madrid seismic zone, where earthquake
depths up to 21 km can be equal to or larger than the epicentral
distances.
Figure 3 shows a schematic of the locator construction and
operation, and figure 4 shows the locator in use. The front of the
display contains a set of educational materials and instructional
activities, a map showing four seismic stations to be used in the
exercises, their seismograms, and a time-distance scale with
adjustable markers. The time-distance scale shows both the
Ts – Tp times and the corresponding distances traveled by the
seismic waves. The distance portion of the time-distance scale is
at the same scale as the map. The Ts – Tp times on the scale are
based on the equation:
678 Seismological Research Letters Volume 77, Number 6 November/December 2006
setup Ts – Tp time and distance
markers on appropriate string
seismic
station
hole in map, string
passes through
hole, behind map
move the union
of the ends of
the strings until
markers line up
(at zero).
T=0
D=0
string passes
over time-distance
scale
spring
loaded
reels
end of each string and the corresponding
marker each move the same distance
hypocenter
d
epicenter
T=0
D=0
Ts – Tp time and distance = v × (Ts – Tp) scale
▲ Figure 3. This figure shows a schematic diagram of both construction and use of the analog earthquake locator. For clarity, only three
stations are shown for the method based on Ts – Tp times. The time and distance scale bar is produced by using the map scale for the distance part of the scale and the corresponding Ts – Tp time as a function of distance based on the velocity model. Both the time and distance
scales are printed on the scale bar. To use the locator, one lets all the strings retract so that the end of each string is at the location of the
seismic station. One then places the marker at the Ts – Tp time for a given station on the string that corresponds to that station (top). Next,
the ends of the strings are joined together and the union is moved until all the markers line up at zero (bottom). As the ends of the strings
are pulled from the locations of the seismic stations, the markers on each string move the same physical distance as the end of the string.
One can determine the distance the end of each string moves using the distance scale. For earthquakes with depths similar to or larger
than their epicentral distances, the union has to be moved noticeably away from the map surface. One can then determine the depth of
the earthquake by measuring the distance from the union of the ends of the strings from the map surface with a ruler that is marked at the
same scale as the map. The epicenter is the point directly below the union.
£ VV ¥
D = ²² p s ´´ Ts < T p
¤V p < V s ¦
D = VT
where D is the length of the ray path, Ts is the S-wave arrival
time, Tp is the P-wave arrival time, Vs is the S-wave velocity, and
Vp is the P-wave velocity. For this implementation of the locator
we used a homogeneous half-space model for the New Madrid
seismic zone with Vp = 6 km/s and
(
V s = 3.5 =
Vp
)
km/s.
3
The map is approximately 0.5 m by 0.5 m and covers a 40 km2
area over the thrust arm of the New Madrid seismic zone. In this
region the seismic stations have a spacing of 10 to 20 km, and the
earthquakes vary from 5 to 21 km in depth. The region includes
Reelfoot Lake, located in the northwest corner of Tennessee,
and a section of the Mississippi River. Holes are drilled through
the map at four seismic stations (shown by stars). At each hole,
a metal loop is attached to a string that can be pulled out of the
hole away from the surface of the map. The other end of the
string passes over the time-distance scale where markers show
the distance the end of the string has moved. Retracting mechanisms mounted to the back of the locator keep the strings taut
and retract the string when the end is moved back toward the
hole. In Volponi’s original analog locator, the map was mounted
on the top of a table. Short sections of pipe, to which the ends
of the strings were attached, were slipped over the four table legs
and provided tension in the strings as the pipes slid up and down
the table legs with movement of the other end of the string.
The components of the locator are very simple, consisting
of a wooden frame and board, a laminated and mounted map,
a scale marked in Ts – Tp time and distance, strings, metallic
loops, key-ring retractors, and a device for holding the ends of
the strings together (figure 4). The markers are placed on the
Seismological Research Letters Volume 77, Number 6 November/December 2006 679
▲ Figure 4. This figure shows the earthquake locator in use, its size, and the presentation of the explanatory material. The four strings
are connected and brought to a point above the map where all the markers line up on the scale at zero time and distance. This point is the
hypocenter. The distance of the union from the map is measured with a ruler that is marked at the same scale as the map. The epicenter
is the point directly under the union where the ruler touches the map.
680 Seismological Research Letters Volume 77, Number 6 November/December 2006
▲ Figure 5. This figure shows the initial placement of the markers at the Ts – Tp times and the dual time and distance scale.
string associated with each seismic station based on the Ts – Tp
time for that station (figure 5). The position of the marker on
the string is also equal to the distance the seismic waves traveled
from the focus to the respective seismic station; this distance
can be read off the distance scale.
The locator can be operated several ways using the Ts – Tp
times. In the first method, the markers for the Ts – Tp arrival
times are placed at the appropriate positions on the scale for
each seismic station, the ends of the strings are pulled together
and joined, and this union is moved around in 3-D until the
Ts – Tp markers line up on the time scale. In most cases, the
union of the ends of the strings must be pulled away from the
map to accomplish this. Because the markers are initially at the
distance the seismic waves traveled to each station, the markers
also line up at zero. When the markers are lined up, the position
of the union represents the earthquake focus and the strings represent the ray paths to each station. The distance of the union
of the strings above the map, obtained using a ruler that has the
same scale as the map, gives the depth of the earthquake. The
point on the map directly beneath the union is the epicenter.
This is the method one would use if the analog earthquake locator were the principal method of locating an earthquake.
The next method is more didactic but still uses the distance traveled by each of the three seismic waves obtained by
conversion of the Ts – Tp arrival times to distance through the
time-distance scale. Start with all the strings retracted so their
ends are at the seismic stations. Next, place the time markers at
the appropriate positions on the strings using the Ts – Tp scale.
Then pull the end of each string away from the station to bring
its individual marker to zero. Now lock or hold down all the
strings. Finally connect the ends of the strings together to find
the solution. Using data from three stations, the solution will
be the unique place where all the strings are taut. As with the
first method, the location will be above the surface of the map,
indicating the depth of the earthquake’s focus and the epicenter
beneath the intersection.
A variation of the previously discussed method, with additional teaching opportunities, is to begin by working with only
one of the locked strings. Demonstrate this string defines a
hemisphere of a given radius by moving the end in all directions above the map while keeping it taut. With data from one
station, the earthquake will therefore lie somewhere on this
hemisphere. A similar hemisphere can also be found below the
map, defining a full sphere. Next, join the ends of two locked
strings keeping in mind that each string defines a sphere. With
the strings taut, the motion of their union above and below
the map is restricted to a circle, defining the intersection of the
two spheres, on which the earthquake is now constrained to lie.
Finally, connecting the ends of all three locked strings together
one finds a single place above the map where all the strings are
taut. This place represents the earthquake focus or hypocentral location. It is much easier to see this intersection using the
strings as radii, rather than trying to draw the intersection of
the three spheres. If more data are available (more seismic stations with locked, measured strings attached), there will still be
only one place where all the strings are taut, but only if the data
are perfect. If the data are not perfect there will be at least one,
and possibly several, places where at least three of the strings
will be taut.
The analog computer can also be used to locate earthquakes using P-wave arrival times only. This is accomplished by
changing the time-distance scale to one showing P-wave travel
times and the corresponding distances P waves travel. Unlike
the case for using Ts – Tp , where the velocity used to scale time
to distance does not represent a physical velocity; the velocity
used to generate the new scale is the actual P-wave velocity. In
this method, one does not know the distance to the hypocenter
beforehand so one cannot lock the strings at a fixed distance or
know where they should line up. The direction of the time and
distance marks on the scale now run the other direction and one
places the time markers based on their relative P-wave arrival
times. The last arrival defines zero time and distance. The earlier arrivals are marked at their relative arrival times/distances
ahead of the last arrival. The ends of the strings are again joined
and the union moved around until the markers line up. In contrast to the case of using Ts – Tp , the markers will now line up at
Seismological Research Letters Volume 77, Number 6 November/December 2006 681
some arbitrary position. The distance from the station with the
latest P-wave arrival time to the earthquake is read directly from
the distance portion of the scale and the distances to the other
stations can be determined by the relative arrival times and corresponding distances. For this method, P arrivals from at least
four seismic stations are required to estimate the four parameters—latitude, longitude, depth, and origin time. (There are
actually six data measurements in the Ts – Tp method as both
P- and S-wave arrival data are needed at each of the three stations. In addition, in the Ts – Tp method we are estimating one
less parameter because the origin time is determined from the
known distance to each station.) The concepts associated with
using only P-wave arrival times are more advanced than those
associated with the Ts – Tp method and would be better suited
for undergraduate or graduate class presentations. An interesting note is that by properly redefining things (stations become
satellites and the earthquake becomes the observation point),
this analog computer can also can be used to demonstrate how
one locates oneself using the global positioning system (GPS).
College-level classes may benefit from demonstrations of
other concepts that are more advanced than those presented
with the museum display. The analog locator can be used to
illustrate the idea of a “best fit” when one has more than the
minimum number of data and introduces noise or errors into
the arrival-time measurements. This can be demonstrated using
four sets of Ts – Tp times. For perfect data, the four markers will
all line up at zero. For data with errors, the user will have to
find a “best” arrangement of the markers near zero. The analog locator can also be used to show how the determination of
origin time and depth are coupled when one has only P-wave
arrival times. As the union is moved vertically over the epicenter, there is very little relative movement between the markers,
so deciding where they best line up is difficult. The device is also
well suited to illustrate problems arising when the earthquake is
well outside the network. In this case, with either type of data
(Ts – Tp or P), one can move the union perpendicular to the
line connecting the union to the stations a considerable amount
with very little variation in the position of the markers.
EXPLANATORY AND EDUCATIONAL MATERIALS
About 2,000 people visit the PERC yearly and the average visitor is between the ages of 8 and 15. The PERC has presentations about paleoseismology, plate tectonics, geology, earthquake hazards—especially in the New Madrid area—and how
CERI records earthquakes in the New Madrid seismic zone.
While several of the activities involve seismology, none are dedicated to the subject of locating earthquakes. The physical analog
model provides a hands-on, interesting, and fun way to obtain a
basic understanding of the earthquake location process.
Explanatory materials are displayed on the locator to give
an introduction to the process of locating earthquakes. Given
the diverse target audience of the PERC and the relatively complex nature of earthquake location, it was important to select
and develop material that was rich in content and suitable for
both adults and children. The goal was to create a model that
would appeal to the average 8- to 15-year-old visitor and also to
visitors that we hope to do a better job of attracting (ages 15 to
24). The introductory materials describe the concept of locating
earthquakes using the travel time of waves, which is analogous
to the common method of estimating the distance to a lightning bolt by counting the seconds between the flash and the
thunder and dividing by five to get miles. Several examples of
seismograms are shown with the P and S arrivals marked. These
seismograms are used to provide the data for the hands-on exercise. Finally, there is a short discussion with a simple explanation of seismic waves, their velocities, the idea of ray paths, and
the relation of the elements of the analog model to the actual
physics of the problem.
Additional materials are available that teachers can use
to prepare their classes before the visit and that teachers and
students can take with them when they leave. Importance was
placed on tying the subject matter to the State of Tennessee science standards. Earth and space science, physical science, geography, and science as inquiry are the main connections to the
curriculum. Academic “bullets” (and grade levels) addressed in
the display include:
• energy (K–12)
• Earth’s features and structure (K–12)
• waves (4–12)
• position and motion of objects (3–12)
• force and motion (K–12)
• measurement skills and tools (K–12)
• map reading (3–12)
• properties of objects and materials (K–12)
• what is scientific inquiry (3–12)
• what abilities are necessary to do scientific inquiry (3–12)
CONSTRUCTION
Creating a functional earthquake locator for use by children
requires considering factors such as cost, durability, accessibility, and portability. Designing the locator to work in the New
Madrid region makes the hands-on exercise more interesting to
local visitors. Using sites from the New Madrid Cooperative
Seismic Network in the region of the seismic zone with the
deepest events ensures that the union will be noticeably above
the map surface. Durability is also a primary concern because
the PERC, like all children’s museums, has a history of broken
displays due to over-enthusiastic use by visitors.
The parts of the display are all relatively inexpensive and
easy to obtain. The base of the display is a sturdy, solid wood
frame with a plywood display board. Tensioning and retracting the strings is performed by a set of key-ring retractors.
Explanatory materials are laminated and mounted on foam
core. The scale, also laminated and mounted on foam core and
with marked strings, is enclosed in a case with a Plexiglas front
panel. The Plexiglas panel can be removed easily to change the
scale to another velocity model or to use P-wave arrival times,
or to change the positions of the markers to use other sets of
Ts – Tp arrival times. The removable panel allows the markers to
be set up beforehand and once reinstalled prevents the markers
682 Seismological Research Letters Volume 77, Number 6 November/December 2006
from being modified by the user. For normal presentations to
grade-school groups, the markers are preset and not accessible.
The space available in the PERC and the desire for portability determined the dimensions of the display. In the PERC,
the locator is mounted so that the base is about three feet
from the floor. The scale is horizontal and positioned above
the map at about eye level so it is easily accessible for gradeschool students.
The strings are sturdy, 20-lb. fishing line with one end connected to the key-ring retractor and the other secured to stainless steel thumb bolts at the seismic station locations. The strings
run behind the locator, come up to the top where they pass over
the time-distance scale, and continue to the key-ring retractors.
The string is orange and has low friction and little elasticity. The
orange color allows the string to be seen by viewers throughout
a classroom or display area. Plastic female banana plugs, with
the electrical interior part removed, keep friction at a minimum
where the strings pass through the holes in the map. Tension,
provided by the key-ring retractors, is necessary to keep the
strings taut and to retract them when the end is moved closer
to its hole. When a string is pulled out from the board, the
appropriate marker moves the same distance horizontally along
the time-distance scale as the distance the end of the string is
moved. The markers are lightweight foam cylinders about 1 cm
in length and diameter. The markers need to be light enough
not to weigh down the string, small enough not to interfere
while moving past one another as the strings are moved, and
large enough to be seen from a distance. Finally, copper wire
was used to fashion a simple connector to hold the thumb bolts
at the ends of the strings together.
CERI makes presentations at schools, science fairs, and
other venues as part of its education and outreach activities.
Demonstrations and hands-on activities are part of these presentations. The locator is well suited for such activities. In the
PERC, the only available space requires a wall-mounted display. When used outside the PERC, the analog locator must be
freestanding and stable as users pull on the strings. The locator
therefore has a support that can be temporarily clamped onto a
table to allow safe operation. The base is quickly set up and easily
disassembled. The display itself is manageable by one person.
It is important that the earthquake used in the exercises
requires that the union of the strings be pulled away from the
map surface to locate the earthquake. Choosing a sub-region
of the network with both dense station spacing and relatively
deep earthquakes ensures that earthquakes with depths greater
than ~5 km will require the union of the strings to be noticeably lifted away from the surface of the map. Finding an example earthquake at such a location and with sufficient depth was
not difficult. Finding an appropriate seismogram, however, was
problematic due to the location of the New Madrid seismic zone
and the seismic network in the Mississippi Embayment, where
the low-velocity unconsolidated sediments of the Mississippi
Embayment lie above high-velocity basement rocks, causing
several complications. The S wave, for example, typically does
not show up on the vertical component seismogram, whereas
a strong S to P conversion from the bottom of the embayment
sediments is evident. Looking at multiple seismogram components and dealing with mode conversions to obtain the P- and
S-wave arrival times is too complex for the didactic purposes
of the locator in the PERC. Selecting a few seismograms that
provide appropriate Ts – Tp values solved this problem.
CONCLUSION
Analog devices provide an excellent format to present complex
physical phenomena in an easy to understand manner that is
often very true to the original physics. The hands-on control of
the locator also makes it a more engaging educational experience. Building the locator provided an interesting undergraduate project that spanned tasks from basic carpentry to elementary education standards to simple seismological theory.
ACKNOWLEDGMENTS
We would like to thank the late Fernando Volponi for showing us the locator in use at the Instituto Geofísico Sismológico
Zonda in San Juan, Argentina, and I. Selwyn Sacks for his advice
about using analog devices to solve problems in geophysics. We
would also like to thank Chris Watson and Ethan Wilding for
help in the early stages of construction, Greg Steiner and David
Steiner for help with design and construction, Kathy Tucker for
development of the map, Tanya Broadbent for help with graphics, and Bill Cupo for photography. We thank Chris TillichWalker, Chuck Langston, Chris Powell, Tanya Broadbent,
Buddy Schweig, Arch Johnston, Michelle Dry, and Susan
Hough for helpful reviews. This work was supported primarily by the Mid-America Earthquake Center of the Earthquake
Engineering Research Centers Program of the National Science
Foundation under Award Number EEC-9701785. We received
additional support through matching funds from the Center
for Earthquake Research and Information of The University of
Memphis. This is CERI contribution number 507.
Center for Earthquake Research and Information
The University of Memphis
3876 Central Ave., Suite 1
Memphis, Tennessee 38152-3050
[email protected]
(R.S.)
[email protected]
(D. S. S.)
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